A Polynomial Kernel for Block Graph Deletion

In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k , and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel wit...

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Bibliographic Details
Published in:Algorithmica Vol. 79; no. 1; pp. 251 - 270
Main Authors: Kim, Eun Jung, Kwon, O-Joung
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2017
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Deletion
Diamonds
Graph theory
Graphs
Kernels
Mathematics of Computing
Theory of Computation
Block graph
Fixed parameter algorithm
Block decomposition
Kernelization algorithm
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k , and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with O ( k 6 ) vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of ‘complete degree’ of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time 10 k · n O ( 1 ) .
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-017-0316-2